Regularized Estimation of High-Dimensional Vector AutoRegressions with Weakly Dependent Innovations

TL;DR

This paper investigates the properties of LASSO estimation in high-dimensional vector autoregressive models with heavy-tailed, weakly dependent innovations, broadening the scope beyond Gaussian assumptions.

Contribution

It establishes oracle properties for LASSO in models with weakly dependent, heavy-tailed innovations under minimal assumptions, extending current high-dimensional time series theory.

Findings

LASSO achieves oracle properties under weak dependence.

Applicable to heavy-tailed innovations with minimal assumptions.

Includes sequences like L^1-NED and strong mixing as special cases.

Abstract

There has been considerable advance in understanding the properties of sparse regularization procedures in high-dimensional models. In time series context, it is mostly restricted to Gaussian autoregressions or mixing sequences. We study oracle properties of LASSO estimation of weakly sparse vector-autoregressive models with heavy tailed, weakly dependent innovations with virtually no assumption on the conditional heteroskedasticity. In contrast to current literature, our innovation process satisfy an $L^1$ mixingale type condition on the centered conditional covariance matrices. This condition covers $L^1$-NED sequences and strong ($\alpha$-) mixing sequences as particular examples.